FOM: Hilbert's 5th Problem

[Torkel Franzen]

 Moshe' Machover writes:

  >I'd be interested to hear FOM contributors' views on the foundational
  >status of nonstandard analysis.

  Probably from most points of view, non-standard analysis is an
interesting development in mathematics that is not "only" technical.
Just to mention two striking aspects, it yields a justification of
something very like the traditional manipulation of infinitesimals,
and it exploits, surprisingly, the limitations of first order logic to
turn pathologies into new and useful mathematical structure.

  So is this a foundational development? That would depend on how we
interpret "foundational". In the sense of foundations put forward by
Steve Simpson, it's not foundational, because it is a mathematical
theory that comes late in the chain of explanation and justification
in mathematics. The basics of non-standard analysis use a lot of stuff
about the real numbers and some logical theory. (Nor is there in
non-standard analysis any uniquely characterized extended domain of
numbers.) We don't set out to explain the real numbers to anybody in
terms of non-standard analysis, and it's far from obvious that this
would be possible.

  Note: I'm not by any means asserting that calculus and the real
numbers cannot be explained and introduced, at the basic level, in
very different terms from those now commonly used. This is done in
intuitionistic mathematics, if nothing else. But non-standard analysis
as it now exists is not a foundational subject.

  The above comments are mainly intended as remarks about "foundational".
I think it's clear enough what Steve has in mind in talking about
foundations (although it would, I think, be a good idea not to invoke
Peikoff, who is rabidly opposed to mathematical logic, in this context).
The foundational concepts and claims are those that we explicitly or
implicitly invoke in introducing, teaching, justifying or explaining
mathematics, and for which we don't present any mathematical justification
or definition.

  In looking through the (very interesting) archives that have already
accumulated for the list, I spotted a reference or two to Kreisel. 
No doubt many of us are familiar with remarks of Kreisel's to the
effect that what is "foundational" in the above sense need not be
fundamental in a more interesting or rewarding sense, and that a
law of diminishing returns is at work in pondering and chewing on
the concepts that first occur to us. However, even if we accept those
remarks it doesn't follow (as Kreisel himself points out) that
traditional foundational studies are necessarily pointless or bound
to lead to nothing of interest.